That’s the argument advocates of direct democracy generally support, but no actual democratic systems fully implement direct democracy. In practice what we generally have is representative democracy, where we delegate the power to make laws, and make and implement policy to representatives.
Those representatives have personal moral responsibility for the laws and decisions they make and are sovereign individuals in their own right, as is any citizen. There’s an argument (I think a strong one) that having been elected they have a right to make laws or rule according to their own beliefs and conscience, within the limits of the law. They’re not just the compliant meat puppets of “the people”.
Of course there are examples of direct democracy, in the form of referendums on specific issues, and these highlight many of the problems with direct democracy. You end up with votes raising spending, alongside votes cutting taxes, with a side order of votes banning government borrowing. You get situations like Switzerland voting to align with the EU along with its free movement provisions, then a referendum mandating ending free movement (but not any other aspects of the treaty), then a referendum confirming maintaining the EU treaty. Take Brexit, what does it mean? Hard Brexit, soft Brexit, there are infinite different possible variations. Which one did the British public intend? Well, different members of the public intended different things, who gets to choose? What happens when the public vote for one thing but most elected people in government responsible for implementing it think it’s a bad idea?
The problem is “the people” don’t have a single coherent group consciousness capable of reconciling competing priorities. That’s what leaders are for.